Metal Process Simulation Laboratory
Department of Mechanical and Industrial Engineering
University of Illinois at Urbana-Champaign
Urbana, IL 61801
CON1D Users Manual
Version 8.0
Brian G. Thomas
Continuous Casting Consortium
Report
Submitted to
Allegheny Ludlum
ARMCO, Inc.
Columbus Stainless
Inland Steel
LTV
Stollberg, Inc.
May 26, 2004
********************************************************************
CON1D User's Manual
Version 8.0
Brian G. Thomas
University of Illinois at Urbana-Champaign
Department of Mechanical and Industrial Engineering
1206 West Green Street,
Urbana, IL 61801
Acknowledgments
The CON1D program has been developed by many different graduate students and
researchers working since 1987 in the Metals Process Simulation Laboratory at the
University of Illinois under the direction of Professor Brian G. Thomas. Major code
developers include Brian G. Thomas, Bryant Ho, Guowei Li, David Stone, and Ya
Meng. Other contributors include Nick Youssef, Avijit Moitra, and Ying Shang. This
project has been sponsored by the Continuous Casting Consortium at UIUC, and the
National Science Foundation (Grants MSS-8957195 and DMI9800274). Ongoing
model calibration and validation has been made possible by experimental data and
measurements on operating casters, which have been provided by several different
steel companies, notably including LTV and Armco.
May 26, 2004.
********************************************************************
2
Part I Introduction
Welcome to CON1D! CON1D is a Fortran program which models heat transfer and
solidification in the mold region of a continuous caster. The model simulates one-dimensional
transient heat transfer-solidification in the steel shell coupled with 2-D steady-state heat conduction
in the mold. Hence, the model is most easily applied to regions away from the corners of the crosssection. It is intended for the study of steel slab casters, but can also be applied to other processes.
The heat flux extracted from the solidified shell surface can either be supplied as a specified
function of distance below the meniscus, or can be calculated using the interfacial model included in
the program. The superheat can be treated in three different ways in the program: 1) calculating
temperature in the liquid steel; 2) supplying a superheat flux profile as a stepwise linear function of
distance below the meniscus; or 3) letting the program calculate the heat flux added to shell surface,
based on previous 3-D turbulent flow calculations. The program can simulate wide/narrow face,
outer/inner face of molds (with or without curvature). It is also capable of calculating heat transfer as
the strand passes by each roll in the spray zones beneath the mold.
A large quantity of information can be obtained using CON1D, which runs in only a few
seconds on a UNIX workstation, such as a Silicon Graphics 4D / 35. The output results include the
following variables (as a function of distance below the meniscus):
(1) Temperatures: mold hot face, cold face, shell surface and shell interior, cooling water
(2) Shell thickness (including positions of liquidus, solidus, and shell isotherms);
(3) Heat flux leaving the shell (across the interfacial mold / shell gap);
(4) Ideal mold taper (based on 1D shrinkage calculations);
(5) Thickness and velocity of solid and liquid flux layers in the mold/shell interfacial gap;
In addition, the model derives important constants: (solidus temperature, liquidus temperature, mean
heat flux in the mold, heat balance at mold exit, maximum temperature in the water channel, negative
strip time, oscillation mark pitch, etc.).
The model warns when there may be boiling in the water channels, if the mold cold-face temperature
exceeds the water boiling point at that pressure. It also indicates when there may be excessive mold
friction, if the lubricating liquid flux layer completely solidifies before mold exit.
Finally, the model outputs the temperatures of mold thermocouples, whose positions are input by the
user. This allows easy calibration of the model with known data, by rerunning CON1D until the
predicted mold temperatures at these positions match the measured ones.
3
The model features user-friendly input of the following comprehensive list of variables affecting
mold heat transfer, (reasonable defaults are provided for many of these):
complete mold geometry including
water slots and bolts
mold curvature
mold platings: variable thickness distribution and conductivity down mold
scale buildup on water channels
mold flux properties and adjustable parameters
temperature-dependent viscosity and conductivity
variable slag rim thickness down mold
powder consumption rate
solid flux velocity
temperature and composition-dependent properties of steel, copper, and water
thermal conductivity
specific heat
density
steel phase (austenite or ferrite), latent heat, solidus and liquidus temperatures
water inlet temperature, velocity, and pressure
oscillation mark size, frequency and stroke
- effect on heat flow
- effect on mold flux consumption
effect of fluid flow on superheat delivery to the shell
- enhanced thermal conductivity in the liquid
- specified superheat flux to the solidifying interface
- model predicted superheat flux (slab casters)
spray zone variables
- hear transfer coefficient Model
- roll space, roll radius, contact angle
- nozzle spray width and length, water flow rate
4
Part II How to run CON1D
(1) How to run CON1D on IBM-PC
(after copying all contents of floppy disk to a single folder on your PC hard drive)
a) Copy sample input data file to 'xxxx.inp' (where xxxx is any 4-character identifier you
want). Edit this file to change the data as desired to match your conditions of interest
(See Part III)
b) Run con1d by typing "con1d" (from a DOS window) or
double clicking on the con1d.exe file
c) Answer the questions asked interactively
d) Examine output data contained in following output files:
xxxx.ech
xxxx.ext
xxxx.frc
xxxx.fxt
xxxx.gpt
xxxx.gpv
xxxx.mld
xxxx.prf
xxxx.pf2
xxxx.prp
xxxx.shl
xxxx.shr
xxxx.spr
xxxx.sst
xxxx.tpr
(echo file containing interactive input data)
(conditions at mold exit)
(phase fraction of shell surface and under certain depth below surface)
(flux temperature distribution)
(gap heat transfer coefficients and film thicknesses)
(consumption check)
(summary of mold temperatures and heat fluxes)
(temperature distribution and heat balance at mold exit)
(temperature distribution and heat balance at user interested point)
(steel properties)
(shell temperatures and taper histories)
(gap shear stress distribution)
(heat transfer coefficients in secondary cooling zones)
(steel shell temperature below surface)
(taper history)
xxxx.liq
xxxx.sol
xxxx.lqi
xxxx.sli
xxxx.seg
(liquid concentration at surface)
(solid concentration at shell surface)
(liquid concentration at some distance under shell surface)
(solid concentration at some distance under shell surface)
(solidification time, CR, SDAS, Tsol from shell surface to inside)
e) Extract data from these files and create graphics using your favorite Personal Computer
or Macintosh graphics programs (eg. Kaleidograph or Excel) OR:
Make plots in a window on your monitor by running the program wgnuplot and
executing the 3 macro-routines provided by typing the following at the prompt in
wgnuplot:
Call 'g.mld' 'xxxx' 'yyyy' (compare mold temperature, shell temperature, and heat flux
results from two runs xxxx.mld and yyyy.mld)
Call 'g.shl' 'xxxx' 'yyyy' (shell thickness results)
Call 'g.gpt' 'xxxx' 'yyyy' (interfacial flux thickness results)
After viewing each plot, type carriage return for next plot.
5
(2) How to run CON1D on workstation
a) Transfer files in sub directory src.unix to workstation
(Warning: check last line to make sure the file transferred correctly)
b) Type "make" to compile and get the executable file 'con1d' on workstation
c) Edit input data file and run CON1D as in (1) a)-d)
d) View data using the 3 gnuplot macros: plotg.mld, plotg.shl or plotg.gpt by typing
at the unix prompt: plotg.mld xxxx yyyy
plotg.shl xxxx yyyy or plotg.gpt xxxx yyyy
Use ^C to get next plot. Alternatively, copy results files to another computer and
postprocess with Kaleidograph or Excel etc..
6
(3) How to calibrate CON1D:
a) Verify that program works OK by running example input file
b) Copy example input file to a new file xxxx.inp (where xxxx is any 4-character
identifier) and edit to match the caster and conditions of interest
c) Run the model until the following match:
1. Mean heat flux in the mold should match measured mold heat flux (based on
performing a heat balance on the cooling water) within about 3%.
(The heat balance calculates the average heat extraction rate (kW/m2) from the top
of the mold to the specified distance, usually mold exit)
Total power extracted per wide face (kW) = qttot (kW/m2) * zmold * W
2. Mold water ∆T for individual water cooling channels (slots) should match just as
well as mean heat flux.
Note that the total water channel cross section area must be input to obtain the
predicted temperature increase compared with measured data.
Change input parameters which are uncertain, and rerun until a match is obtained.
Typical variables to change include: mold flux properties, (conductivity,
emissivity) air gap thickness profile down mold, and speed ratio of the solid flux.
d)
After getting mold water / heat balance to agree, compare predicted and measured
mold temperatures at thermocouple locations. Model temperatures should be
slightly low, if the thermocouples are positioned close to either the water or to hot
bolts (where a slot is missing). This geometric dependency should be checked
with results from a FEA analysis of the mold in order to quantify the offset
distance needed to match the model predictions with results at the thermocouple
location for a known heat flux.
If model temperatures are still low, there may be scale build-up on the mold cold
face. Add a scale layer and rerun the model to obtain a match. (The heat flux
should not change much) If model temperatures are very high, either a) the
thermocouples may have a contact resistance and not be measuring the actual mold
temperature, or b) there is incipient boiling increasing the effective heat transfer
coefficient.
e)
Shell thickness profile down mold should match that obtained from breakout shells
to the extent that the input conditions match the breakout conditions. Timedependent casting speed and precise determination of the time – distance
relationship for the breakout shell is needed when making this comparison.
f)
Run the calibrated model for other input conditions, as desired.
7
3a) CON1D Subprograms
con1d
main program
initialize variables, handle input and output files,
control simulation
ckseg
use Clyne-Kurz simple analysis segregation Model
to calculate liquidus and solidus (optional)
bound
function to determine heat flux leaving shell
(incorporates model of mold/shell interface)
(calls slagthick and check etc. subroutines for calculation)
fluxtemp
calculate temperature distribution in mold flux vs. time & distance
hbal
perform approximate heat balance on shell at specified distance
(usually mold exit)
output temperature distribution and heat balance
compare heat loss from shell surface with heat content in shell:
1) superheat or heat flux to shell inside
2) latent heat
3) sensible heat
input
input and “echo” write xxxx.ech file (using SI units)
(uses functions doutrf and dinnrf to calculate curved mold thickness)
ma
2-D mold temperature calculation
(uses function fyy in calculating y direction)
moldtemp
function to calculate mold hot face temperatures
(based on 1-D, 2-D, or data points)
once
define heat flux to inside of shell (based on user-specified data points)
onceff
define default superheat flux to inside of shell
(based on previous 3-D turbulent flow results)
phase1
determine fraction of α, γ, δ and liquid based on phase diagram
(uses function ae3c to determine Ae3 temperature with carbon)
tle, rk, cp
functions to define temperature dependent properties (TLE, k, Cp)
based on stainless steel or pure iron (props1.f)
para, fracti
composition-dependent property calculation (phase2.f)
define temperature, carbon- and phase- dependent properties for low-carbon
steels
(para calculates coefficients for phase diagram change lines
fracti defines phase fractions for TLE, Cp, k)
print
interpolate between nodal temperatures to locate solidus and liquidus;
calculate ideal taper from simple shell shrinkage (using subroutine tpr)
output results at specified time interval
tapering
calculate taper according to Dipenaar's method
shearstress
calculate shear stress distribution vs.time
simul
calculate temperatures (shell) at next time step
usertle,userrk,usercp,usertemp
define TLE, k, Cp, liquidus, solidus by user
waterh
calculate heat transfer coefficients between water and mold cold face
These *.for files are compiled into *.obj files and linked using any FORTRAN compiler (eg.
Microsoft 8087) on IBM-PC or standard FORTRAN-77 on UNIX workstations.
8
(3b) List of Unix files contained in CON1D
con1d.f:
bound.f:
program con1d
real function bound(ts,zmm,zmoldmm,tmold,hwater,ispray,inaro,vctmms,
+
tcold1,ibound,iz,twater,dzcm,tsol,freqz,qcold,thotc,zpp,ratio)
subroutine rcalc(rhoflux,rindex,wcao,wsio2,wmgo,wna2o,
+
wk2o,wfeo,wfe2o3,wnio,wmno,wcr2o3,wal2o3,wtio2)
subroutine slagthick(mu0,expn,rhoflux,eslag,esteel, rindex,acoeff,ts,
+
tcrystal,tmold,tkliquid,tksolid,qosc,qcons,vsolid,
+
vctms,dliquid,dsolid,deff,rcontact)
subroutine balanc (a,n,np)
subroutine zrhqr (a,m,rtr,rti)
subroutine hqr(a,n,np,wr,wi)
subroutine check zm,dosc,vctms,t,vsolid,dsolid,dliquid,rhoflux,
+
expn,mu0,vosc,freqz)
function liqthick(tmpdliq,mu0,expn,rhoflux,vctms,qcons,qosc)
function oscthick(pitch,dliquid,dsolid,tkeff,zm,ioscflag)
ckseg.f:
subroutine ckseg(tckliq,tcksol,tckperi,SDAS,CR)
subroutine segoutput(time,temp,iFileHandle1,iFileHandle2, zmm1,zmm2)
fluxtemp.f:
subroutine fluxtemp_time(dzcm,vctmms,ratio)
subroutine fluxtemp(zmm,tn)
hbal.f:
subroutine hbal(tnext,qtin,qts,n,iout1)
input.f:
subroutine input(ibatch,nconsf,ibound,i2d,pwater,nflux,nthc)
real function doutrf(dmld1,amrad,zm,zmold,dmen,dwater)
real function dinnrf(dmld1a,amrada,zm,zmold,dmen,dwater)
ma.f:
subroutine ma(z1mm,nz,nthc1,nz0,dmen,xmold)
function fyy(x,y,zmm,c1,fb1,xmold,zmold)
moldtemp.f:
real function moldtemp(qtot,hwater,zm,zmold,tcold,inaro,twater,thotc,
+
dtcoat,dtcupp,number)
once.f:
subroutine once(qfill,max,nflux)
onceff.f:
subroutine onceff(qfill,max)
phase1.f:
subroutine phase1(tliq,tsol,tperi,carbeq,pc,ae1,ae3)
function ae3c(carbon)
phase2.f:
subroutine para
subroutine fracti(tc,pc)
print.f:
subroutine print(n,t,printflag)
subroutine tpr(tsurf,def,n,t)
props1.f:
function tle(tc,pc)
real function rk(tc,dkdt)
real function cp(tc)
shear.f:
subroutine shearstress(vctmms,ratio)
simul.f:
subroutine simul (qfill,t,tnext,q,n,maxq)
tapering.f
subroutine tapering(t,tnext,shrink,zmm,dzcm,zmold,tsol,tliq,pc,n)
user.f:
subroutine userrk(tc,dkdt,rk)
subroutine usercp (tc,cp)
subroutine usertemp (tliq,tsol,tperi)
subroutine usertle (tc,pc,tle)
waterh.f:
subroutine waterh(zm,zpp,hwater,tcold,twater,tkmold)
9
Part III Theory
III-1.1 Heat Conduction in Solidifying Steel Shell
Temperature in the solidifying steel shell is governed by the 1D transient heat conduction equation :
2
∂ 2T ∂ k  ∂ T 
∂T
*
= k steel 2 + steel 
ρ steel Cpsteel
[3.1]
∂x
∂t
∂ T  ∂ x 
where Cp* , the effective specific heat for the solidifying steel, is defined as:
df
dH
= Cp − Lf s
C *p =
dT
dT
[3.2]
The simulation domain, a slice through the liquid steel and solid shell, together with the
boundary conditions, is presented in Figure 3.1.
mold
interface
shell
simulation domain
solid shell
n
∆x
liquid steel
i
Figure 3.1 The simulation domain in steel
10
1
Equation [3.1] is solved at each time step using the following explicit finite-difference
discretization (a central difference scheme):
(1)
Interior nodes:
Ti new = Ti +
(2)
[3.4]
Liquid boundary node (adiabatic boundary condition):
T1new = T1 +
(3)
∆t ⋅ k
∆t
∂k
2
T − 2Ti + Ti +1 ) +
(Ti +1 − Ti −1 )
2
* ( i −1
2
*
∆x ρ Cp
4∆x ρ Cp ∂T
2∆t ⋅ k
(T2 − T1 )
∆x 2 ρ Cp*
[3.5a]
Shell surface node (with heat flux boundary condition):
2
new
n
T
2∆t ⋅ k
∆t ∂k  qint  2∆t ⋅ qint
= Tn + 2
T − Tn ) +
−
* ( n −1
∆x ρ Cp
ρ Cp* ∂T  k  ∆x ρ Cp*
[3.5b]
The shell surface temperature is denoted as Ts:
Ts = Tnnew
where qint = hgap (Ts − Thot ) , and Thot is the mold hot face temperature, which can be obtained
together with qint and hgap through iteration. The details are presented in section III.
(4)
Steel thermal properties (Tliq, Tsol, ρ, Cp, k and Lf):
Tliq and Tsol, are calculated by the program as a function of steel composition, based on the
phase diagram for low-alloy steel. The properties, ρ, Cp, k and Lf can be treated in three ways. First,
the carbon content and temperature dependent properties can be calculated based on the phase
diagram for low carbon steel by the program (select option icp = 1000 in input data file, see Part IV).
Alternatively, temperature dependent properties can be found from a set of empirical formula for
stainless steel. Finally, ρ, Cp, k and Lf can be input as constants.
III-1.2 Superheat Flux
Before it solidifies, the steel must first cool from its initial pour temperature to the liquidus
temperature. Due to turbulent convection in the liquid pool, the “superheat” contained in this liquid
is not distributed uniformly. A small database of results from a 3-D fluid flow model3 is used to
11
determine the heat flux delivered to the solid / liquid interface due to the superheat dissipation, as a
function of distance below the meniscus, qsh. Examples of this function are included in Fig. 3.2,
which represents results for a typical bifurcated, downward-directing nozzle. The initial condition on
the liquid steel at the meniscus is then simply the liquidus temperature.
This superheat function incorporates the variation in superheat flux according to the superheat
temperature difference, ∆Ts, casting speed, Vc, and nozzle configuration. The influence of this
function is insignificant to shell growth on most of the wide face, where superheat flux is small and
contact with the mold is good.
For the first node, where temperature is lower than liquidus temperature, the effect of
superheat on node temperature is:
Ti = T i +
∆t
q
ρ C p∗dx sh
[3.6]
where dx=∆x for interior nodes, and dx=∆x/2 for boundary nodes.
Heat flux (MW/m2)
4
3
2
1
top of
mold
Peak near
meniscus
400
Solidifying
Steel Shell
800
Narrow face
(Peak near mold exit)
Heat Input
to shell inside
from liquid
(q )
sh
Liquid Steel
Mold Exit
Water Spray Zone
z
Fig. 3.2.
Meniscus
200
Heat Removed
from shell surface
to gap
(qint)
600
Copper Mold
and Gap
1.0
Wide face
(Very little superheat )
1500 ÞC
1400 ÞC
1300 ÞC
1200 ÞC
1100 ÞC
Distance below meniscus (mm)
x
0
Heat flux (MW/m2)
23 mm
Model of solidifying steel shell showing typical isotherms and heat flux
conditions
12
III-2. Heat Conduction in the Mold
Temperatures within the mold, including in particular the hot and cold face temperatures, are
calculated knowing the heat fluxes, qcold and qint, and the effective heat transfer coefficient to the
water, hwater.
Two dimensional, steady state temperature distribution within a rectangular section through
the mold is calculated in the upper portion of the mold. It is found that the temperature is nearly
linear in the mold thickness direction about 50 mm or so below the meniscus. Thus, a 1D assumption
is adopted in the model below the distance zana below the meniscus.
Tcold
zmen
z
water
channel
zana
zmold
x
Twater
kwater
pwater
Cpwater
q=0
coating layers
meniscus
2D model zone
zpeak
mold
solidifying steel
q int
1D model zone
Thot
Tcold
q=0
xmold = dm
Figure 3.3A Mold Temperature Calculation
13
III-2-1. Effective Heat Transfer Coefficient at Mold Cold Face:
Tcold
Thot
bolt
wch
hot
steel
fin
coating layers
solid mold
Lch
water channel copper mold
twater
bolt
xmold
dm1
dch
Figure 3.3B Water Channel in the mold
The effective heat transfer coefficient between the cooling water and the cold face (“waterside”) of the mold copper, hwater, is calculated using the following formula, which includes a possible
resistance due to scale build-up:
d
1 
hwater = 1  scale +
 kscale h fin 


[3.7]
To account for the complex nature of heat flow in the undiscretized width direction, the heat transfer
coefficient between the mold cold face and cooling water, hfin, is obtained using the following
formula which treats the sides of the water channels as heat-transfer fins.
h fin =
2hw kmold ( Lch − wch )
2hw d ch2
hw wch
+
tanh
Lch
Lch
kmold ( Lch − wch )
[3.8]
Where, Lch, wch, dm1, dch are geometry parameters shown in Figure 3 and km is the mold (copper)
thermal conductivity. The presence of the water slots can either enhance or diminish the heat transfer
relative to a mold with constant thickness (equal to the minimum distance between the root of the
14
water channel and the hot face). Deep, closely-spaced slots augment the heat transfer coefficient,
(hfin larger than hw) while shallow, widely-spaced slots inhibit heat transfer. In most molds, hfin and
hw are very close.
In Eq. 3.8, the heat transfer coefficient between the water and the sides of the water channels,
hw, is calculated assuming turbulent flow through a pipe: [C. A. Sleicher and M. W. Rouse, Int. J.
Heat Mass Transf. V. 18, pp. 677-683, 1975]
hw =
k water
c2
1
5 + 0.015 Recwaterf
Prwaterw
D
(
)
[3.9]
Here D is the equivalent diameter of the water channel, c1 and c2 are the empirical constants.
D=
2wch dch
wch + d ch
[3.10]
c1 = 0.88 − 0.24 ( 4 + Prwaterw )
[3.11]
c2 = 0.333 + 0.5e−0.6Prwaterw
[3.12]
Re f =
Prw =
ρ water vwater D
µ waterf
[3.13]
µ waterwCpwater
[3.14]
kwaterw
The properties of the water, needed in the above equations, can be treated as either constants or
temperature dependent variables evaluated at the film temperature (half-way between the water and
mold wall temperature), according to the selection in the input data file made by the user.
III-2-2. 1D Steady State Temperature Model of Mold:
Hot face temperatures at or near the mold surface are calculated from the thickness of the copper, dm,
the water side heat transfer coefficient, and the interfacial heat flux, explained in a later section.
 1
d 
Thotc = Twater + qint 
+ m
 hwater km 
[3.15a]
15
Further hot face temperatures are calculated by incorporating the resistances of the various thin
coating layers, (Ni, Cr, air gap etc.), which vary with distance down the mold according to the input
file:
 1
d poly
d
d
d
+ m + ni + cr +
Thot = Twater + qint 
 hwater km kni kcr k poly




 1
d poly d air
d
d
d
Tmprime = Twater + qint 
+
+ m + ni + cr +
 hwater km kni kcr k poly kair

[3.15b]



[3.15c]
The hot face temperatures include the surface of the copper, Thotc, the surface of the outermost mold
plating layer, Thot and the temperature at the interface between the air gap and the solid mold flux
layer, Tmprime, including any contact resistance that might be present.
In these equations, the copper thickness, dm, varies with distance down the mold, according to the
mold curvature:
Outer radius:
outer
outer
= d moldo
+ RO 2 −
d mold
2
2
1
1
2
Z mold _ total ) − RO 2 − ( Z mold _ total ) − Z mold
(
_ total
4
4
[3.16]
2
2
1
1
2
Z mold _ total ) + RI 2 − ( Z mold _ total ) − Z mold
(
_ total
4
4
[3.17]
Inner radius
inner
inner
d mold
= d moldo
− RI 2 −
where dmoldo is the mold thickness at the top of the mold, Zmold_total is the total mold length (sum of
working mold length Zmold and distance of meniscus from top of the mold Zmen) and RO, RI are mold
outer and inner radius of curvature respectively.
Tcold is the temperature of the root of the water channel, at the interface between the mold copper and
the scale layer, if present:
Tcold = Twater +
qint
hwater
[3.18]
16
III-2-3. 2D Steady-State Temperature Model of Mold:
By assuming constant thermal conductivity in the upper mold and constant heat transfer
coefficient between the mold cold face and the water channel along the casting direction, the two
dimensional steady state heat conduction equation for mold temperature modification in meniscus
region is the following Laplace equation:
∂ 2T ∂ 2T
+
=0
∂x 2
∂z 2
[3.19]
The analytical solution to this equation with the boundary conditions shown in Figure 3.2 is a
cosine series:
k
 ∞
T ( x, z ) = Twater + c0  mold + x  + ∑ c2 n cos ( λ z ) ( c1e − λ x + e − λ x ) λ = nπ λ = nπ [3.20]
Z2D
Z2D
 hwater
 n =1
(
)
where x is distance through the thickness of the mold, measured from the root of the water slot. The
constants λ, c0, c1 and c2n depend on the heat flux calculated to enter the mold hot face qint, thermal
conductivity km, the effective heat transfer coefficient hfin and the mold geometry.
λ = nπ
[3.21]
Z2D
c0 =

2 ⋅ ∆z  z j +1 + z j
+ bj 
aj
2
Z2D 

[3.22]
c1 =
kmold λ − hwater
kmold λ + hwater
[3.23]
2
2 1
 
Z2D  λ 
c2 n = B λ x
e 1 − c1e −2 λ x
(
)
 ( a j z j +1 + b j ) sin ( λ z j +1 ) − ( a j z j + b j ) sin ( λ z j ) 


B = ∑ a

j
j
cos ( λ z j +1 ) − cos ( λ z j )
+

 λ

(
[3.24]
)
where aj and bj are the linear interpolation coefficients of the interface heat flux in zone j:
from qaj = km (aj zaj + bj) and qbj = km (aj zbj + bj), the aj and bj can be obtained:
17
aj =
bj =
qint j +1 − qint j
kmold ( z j +1 − z j )
[3.25]
qint j z j +1 − qint j +1 z j
kmold ( z j +1 − z j )
The actual hot face temperature of the mold is adjusted to account for the possible presence of mold
coatings and air gaps:
d
d poly d air
d
+
Tmold = T ( d mold , z ) + qint  ni + cr +
 kni kcr k poly kair

18



[3.26]
III-3.1 Heat Flux Across the Interfacial Gap
Heat flux extraction from the steel is governed primarily by heat conduction across the
interfacial gap, whose thermal resistance is determined by the thermal properties and thicknesses of
the solid and liquid powder layers, in addition to the contact resistances at the flux / shell and flux /
mold interfaces and powder porosity, which are incorporated together into a single equivalent air gap,
dair. Non-uniformities in the flatness of the shell surface, represented by the oscillation marks, have
an important effect on the local thermal resistance, and are incorporated into the model through the
depth and width of the oscillation marks. This is used to calculated an effective average depth of the
marks relative to heat flow, deff. The oscillation marks can be filled with either flux or air, depending
on the local shell temperature. When the gap is large, significant heat is transferred by radiation
across the semi-transparent flux layer. This model for gap heat conduction is illustrated in Figure 3.4
and 3.5 and given by the following equation:
solid
flux
mold
Thotc
Tfsol
T'm
temperature
profile
liquid
flux
T's
Ts
shell
equivalent
layer for
oscillation marks
velocity
profile
Vs
x
Vc
ks
dssol
Fig. 3.4
kl koeff
dlliq doeff
Velocity and temperature profiles assumed across interfacial gap
19
qint = hgap (Ts − Tmold )
1
hgap =

d
d
 rcontact + air + solid
kair k solid

hrad
[3.27]



 
1
 + 
1
 
+ hrad
 dliquid d eff
+
 k
keff
liquid

2
2

m 2σ (Ts + Tmold )(Ts + Tmold )

1
1
 0.75a (d
+
−1
liquid + d eff ) +

emold esteel

= 
 m 2σ (Ts 2 + Tcrystal 2 )(Ts + Tcrystal )

1
1
 0.75a (d
+
−1
liquid + d eff ) +

eslag esteel
[3.28]









(T
mprime
(T
crystal
≥ Tcrystal )
< Tmprime )
where
qint = heat flux transferred across gap (Wm-2)
hgap = effective heat transfer coefficient across the gap (Wm-2K-1)
hrad =radiation effective h (Wm-2K-1)
Ts = surface temperature of the steel shell (°C)
Tmprime = mold temperature+mold/slag contact resistance delta T (°C)
Tmold = surface temperature of the mold (outermost coating layer) (°C)
Tcrystal = mold flux crystallization temperature (°C)
rcontact = flux/mold contact resistance(m^2K/W)
dair, dsolid, dliquid, deff = thickness of the air gap, solid, liquid flux, and oscillation
mark layers (mm)
kair, ksolid, kliquid, keff = conductivity of the air gap, solid, liquid flux, and oscillation
mark layers (W/mK)
m = flux refractive index
σ = Stefan Boltzman constant (Wm-2K-4)
a = flux absorption coefficient (m-1)
εs, εm = steel, mold surface emmisitivities
The calculation of some of the above variables is explained in the next section. Other variables are
input data, defined in the Nomenclature section.
20
Tmold
rcontact
dair
kair
dsolid
ksolid
Tmprime ≥ Tcrystal
dliquid
kliquid
deff
keff
Ts ≥ Tsol
1/hrad
Figure 3.5 Thermal resistances used in the interface model
21
Ts
III-3-2. Mass and Momentum Balances on the flux within the interfacial gap
Flux is assumed to flow down the gap as two distinct layers: solid and liquid. The solid layer
is assumed to move at a constant velocity, Vs, which is always greater than zero and less than the
casting speed according to the input factor, fv.
Vs = f v ⋅ Vc
(0 <
f < 1)
[3.29]
The casting speed is imposed at the point of contact between the shell and the liquid layer, which is
assumed to flow in a laminar manner, owing to its high viscosity. Flow in the liquid layer is given by
the Navier-Stokes equation:
∂τ xz
= ( ρ steel − ρ slag ) g
∂x
[3.30]
where x = in the direction across the gap
ρslag = average density of flux
g = gravity acceleration
This formulation ignores axial pressure gradients in the flux channel, which may be generated in the
central regions of the wide face and could be accounted for by replacing -ρflux with (ρFe - ρflux ) in
Eq. 3.30. The effect was found to be negligible, however, except in the vicinity at the meniscus,
where oscillation plays an even more important role.
The tangential shear stress, τxz, is related to the viscosity of the molten flux, µ, by:
τ xz = µ
∂Vz
∂x
[3.31]
The viscosity, µ, is assumed to vary exponentially with distance across the gap, according to
the temperature:
 T −T
µ = µo  o fsol
 T − T fsol




n
[3.32]
where Tfsol is the solidification temperature of the flux, µo is the viscosity measured at T1300, and n is
an empirical constant chosen to fit all of the measured data. Thus the viscosity evaluated at surface
temperature Ts is:
 T −T
µ s = µo  1300 fsol
 T − T fsol




n
[3.33]
22
Mass balance was imposed to express the fact that the known powder consumption must equal
the total powder flow rate past every location down the interfacial gap. The following equation
expresses this condition as a balance on the total consumption, Qf (kg m-2) as the density was
assumed to be constant:
Q f × Vc
ρ flux
= Vsolid d solid + Vliquid dliquid + Vc d osc
[3.34]
where the average depth of the oscillation marks (relative to their volume to carry flux), dosc, is
calculated from:
0.5 Lmark d mark
d osc =
Lpitch
L pitch =
[3.35]
Vc
freq
[3.36]
Equations [3.29], [3.30], [3.31] and [3.33] yield a velocity distribution across the flux layers,
which is illustrated in Fig. 3.6. Integrating across the liquid region yields an average velocity for the
liquid layer, Vl:
V1
where
(ρ
=
flux
2
− ρ steel ) gd liquid
µ s ( n + 2 ) ( n + 3)
2
+
Vc + Vs ( n + 1)
( n + 2)
[3.37]
Vl = avg. velocity of liquid flux (m s-1)
Vc = casting speed (m s-1)
Vs = velocity of solid flux (m s-1)
n = flux viscosity exponent
ρFe = steel density (kg m-3)
ρflux = flux density (kg m-3)
g = gravity (9.81 m s-2)
dl = thickness of liquid flux layer (m)
µ = flux viscosity (Pa-s)
Ts′= steel outside surface temp. (˚C)
Equation [3.34] and [3.37] were solved simultaneously for ds and dl. Special cases arise when
ds = 0 (zone I) or Qf < Vc dosc (zone III). In the former case, (zone I), the solid rim thickness, (via
zrim1 and zrim2) must be input, as it is unaffected by the flux consumption rate. In the latter case,
(zone III), the oscillation mark volume is replaced with air.
The average effective thickness of the oscillation marks, relative to its effect on heat transfer,
is calculated from:
0.5 Lmark d mark
[3.38]
d eff =

k gap 
d mark
( Lpitch − Lmark ) 1 + 0.5 d + d k  + Lmark
liquid
solid
mark 

23
Flux
Steel
doeff
Lpitch
dosc
Lmark
d mark
T = Ts
Figure 3.6 Model treatment of oscillation marks
III-4. Mold cooling water temperature rise:
A heat balance in the mold calculates the total heat extracted, qtin, based on the increments of heat
flux found in the interfacial heat flow calculation, qint (W/m2):
qtin (kJ/m2) =
∑ qint
* ∆t
[3.39]
mold
The mean heat flux in the mold is calculated at the specified distance below the meniscus (usually at
mold exit)
qtin Vc
qttot (kW/m2) = z
[3.40]
The temperature rise of the cooling water is determined by:
∆Tcooling water =
∑
qint Vc ∆t Lch
ρwater Cpwater Vwater (m/s) wch dch
[3.41]
mold
This relation assumes that the cooling water slots have uniform dimensions, wch and dch, and
spacing, Lch. Heat entering the hot face (between two water channels) is assumed to pass entirely
through the mold to heat the water flowing through the cooling channels. This calculation must be
modified to account for missing slots due to bolts or water slots which are beyond the slab width, so
do not participate in heat extraction. So the modified cooling water temperature rise is:
∆Tmodified cooling water = ∆Tcooling water ∗
wch d ch slab width
∗
Lch
totcharea
[3.42]
This cooling water temperature rise prediction is useful for calibration of the model with an operating
caster.
24
III-5. Mold Taper calculation
Previous work involving coupled thermal-stress calculations has investigated the shrinkage
profile of the shell down the mold. This calculation decomposed the total strain into the sum of three
components: the thermal contraction, the plastic strain, and the elastic strain. The first of these has
been found to dominate. Fortuitously, the thermal strain at the surface of the shell has been found to
match quite closely with the average total strain across the thickness of the shell, which controls its
shrinkage. This is because this outer layer of steel solidifies first and shrinks relatively stress free,
while the later, inner steel to solidify against it is weaker and accommodating. Thus, the entire,
complex mechanical behavior of the shell can be approximated quite reasonably by the following
simple calculation for the shrinkage of the shell:
Old Model:
∆W = (TLE (Tsol ) − TLE (Ts ))
W
2
[3.43]
A new model to calculate the shrinkage is based on Chandra's method[1]. The thermal strain is
computed according to the average thermal linear expansion of the solid shell between the two
consecutive time step.
New Model:
 1
W
∆W =   ∑ ( TLE(Tnext(i)) − TLE(T(i)))
 i  i =solid nodes
2
2 ∆W
ZW
2 ∆W
cumulative taper (% per mold) = W
cumulative taper (% per m) =
[3.45]
[3.46]
W + 2 ∆W
Z mold
∆W
[3.44]
W
Figure 3.7 Mold Taper Calculation
25
Ideal taper at mold exit:
2 ∆W
ideal taper (% per m) = z
mold W
ideal taper (% per mold) =
[3.47]
2 ∆W at mold exit
W
[3.48]
where:
ρ
ρ0
Tsol
= density (kg/m3)
= density at reference temperature, To (kg/m3)
= steel solidus temperature
W
∆W
zmold
Z
= strand width between narrow faces
= change in steel shell width (mm)
= working mold length
= z + zmen
TLE is the thermal linear expansion function for the given steel grade,
 ρ0  1/3

defined by TLE (T) = 
ρ(T)
TLE is calculated from weighted averages of the phases present:
TLE (T) = %δ TLE δ(T, %C) + %γ TLE γ(T, %C)
[3.49]
[3.50]
III-6. Heat Transfer in the spray cooling zone:
Heat transfer in the spray cooling zones tracks the steel shell as it moves past each individual
roll. Heat transfer is governed mainly by the spray heat transfer coefficient, which is calculated from
the spray water flow rate (Qw) at each spray zone, using a general equation of the following form:
hspray = A·c·Qwn (1- bT0)
[3.51]
Where, T0 is water and ambient temperature in spray zone. In the model of Nozaki et al.[2],
A·c=0.3925, n=0.55, b=0.0075. Users can choose other models[3] by setting coefficients A, n and b.
Heat transfer is enhanced by nucleate film boiling if the steel shell surface temperature drops below
550˚C.
Heat extraction is a maximum directly beneath the spray nozzle (assumed centered between
the rolls) and at the roll / shell contact region. The relative size of these maxima is governed by the
fraction of heat specified to leave via the rolls relative to that removed by the spray cooling water.
The output including the variation in heat transfer coefficient is given in xxxx.spr.
26
mold exit =
start of zone 1
( 1 roll, irollsw=1)
start of zone 2
(irollsw=2)
mold
zone 1
1
2
zone 2
start of zone 3
(irollsw=3)
2
3
zone 3
3
3
start of zone 4
rollradw
4
zone 4
strand
Figure 3.8 Schematic of a typical spray zone configuration below the mold
27
Part IV Input data:
The input file should be put in a file 'xxxx.inp'. The data included in this file can be written in
free format. The rest of section defines the input variables and their corresponding line numbers in
the input file. Units are specified in both the input data files and in the Nomenclature section, so are
omitted here.
Section (1) Casting conditions:
Variables
in program
nvc
vctime
vcmmin
Comments
number of time-cast speed data points
(if=1, constant speed)
time
casting speed
It is used
(1) to calculate time step
(2) to calculate increment in casting direction
(3) in calculation of negative casting time
(4) to output the heat into the shell
tinit
pour temperature
xslabmm
slab thickness
It is employed
(1) as the default simulation domain
(2) in mold taper calculation
(3) to convert mold powder consumption unit between
kg(flux)/m2 and kg(flux)/tonne(steel)
Line No.
in file
7
10
11
12
13
yslabmm
dmen
slab width (see xslab)
distance of meniscus from top of mold, zmen (see Fig 4.2)
zmoldmm
working mold length (distance between meniscus and mold exit),mm 16
( zmold=zmoldmm, zmoldm=zmoldmm/1000 )
z position for outputting heat balance information in the mold
(This value should be smaller than working mold length.)
17
nozzle submergence depth (meniscus to top of port)
it is used to determine the superheat fluxes
18
zhbal
submerg
28
14
15
Section (2) Simulation parameters:
Variables
in program
Comments
inaro
flag to specify wide face/narrow face simulation
for wide face, meaning a mold
(1) with a curvature mold hot face
(2) including water channels
(3) with spray zones containing rolls below mold
(4) using slab thickness in taper calculation
(5) using wide face default superheat flux
=0
=1
Line No.
in file
21
for narrow face, meaning a mold
(1) with straight hot face
(2) without water channels
(3) with only one spray zone(no rolls) below the mold
(4) using slab width in taper calculation
(5) using narrow face default superheat flux
imoldflq1
=0
=1
=2
type of the mold
slab
funnel mold
billet (oil lubrication)
imoldflq2
flag to specify outer face/inner face
23
Affects thickness variation in vertical direction for molds with curvature
outer wide face
inner wide face
straight wide face or narrow face (no curvature)
=0
=1
=2
ibound
=0
22
flag to specify treatment of mold/shell interface
24
to model heat flux across the interfacial gap between mold and shell,
using input mold flux properties and parameters. This means that the
shell solidification is coupled with mold temperature simulation and flux
motion, so iteration is required.
29
Variables
in program
=2
=1
=-1
Comments
Line No.
in file
to model heat flux across the interfacial gap between mold and shell for
oil casting case, only contact resistance, oscillation mark and air gap are
taken into account.
In these two case, the input data (nbdata, zbdata, bdata) following in
lines 26-29 are ignored.
to enter interface heat flux data (lines 25-28) between the shell and mold as a
function of the distance below the meniscus and calculate solidification of the
shell. (mold is not simulated) (The pc version may have problems with this
option)
to enter interface heat flux data (lines 26-29) between the shell and mold
as a function of the distance below the meniscus and calculate both the
temperature in the mold and solidification of the shell.
nbdata
zbdata
bdata
number of input heat flux data points ( < 20) (see ibound)
z data (distance below meniscus) (see ibound)
q data (interface heat flux) (see ibound)
26
28
29
fflux
flag for treatment of superheat
for calculating temperature in liquid steel to get the heat flux
added to shell surface
In this case the superheat flux input data (in line 32-35) are inactive
30
=0
nflux
zqdata
=1
to use default superheat flux data
It means that the program will create superheat flux data
according to superheat (pour temperature - liquidus
temperature), nozzle submergence, and the simulation
face (wide or narrow). In this case the superheat flux
input data (in line 31-34) is ignored.
=-1
to enter superheat flux data
It means the program will use the user input superheat flux data in line 31-34.
number of input superheat flux data points ( < 20) (see fflux)
z data (distance down mold below meniscus) (see fflux)
30
32
34
qdata
q data (superheat flux) (see fflux)
31
35
Variables
in program
Comments
i2d
flag to choose more accurate calculation in meniscus region or not
for 1D calculation of mold heat transfer, using approximate method
meniscus region
for more accurate 2D mold heat transfer simulation in meniscus
region
for more accurate 2D mold heat transfer simulation for whole mold
(one extra loop for better taper calculation)
36
zana
Maximum distance below meniscus for 2D simulation in the mold
38
dtime
time step size
40
=0
=1
=2
Line No.
in file
The program adopts explicit solution scheme, so a large time
increment will cause solution to be more unstable: dtime should
satisfy the criterion: dtime < 0.5 * ∆x * ∆x * dense *Cp / k
dtime depends on distance increment ∆x (see nstep). The
smaller the distance increment, the smaller dtime should be.
The time step must also be bigger than a lower limit which depends on
the memory limit of the machine, or an error message appears.
nstep
number of slab sections ( < 300)
It is used to determine the increment of distance
(∆x= thmax /nstep,
x node = 1 is at the center of slab,
x node = nstep+1 is at shell surface)
41
freqz
zpmin
printout interval
the z-coordinate (casting direction) of starting output
(z=0 is at meniscus)
42
43
zmax
thmax
maximum simulation length
maximum simulation thickness
(0 = default . The program will use half of slab thickness
xslab / 2 for wide face simulation; and half of slab width
yslab / 2 for narrow face simulation.
44
45
itmax
nshlt
maximum number of time steps
Shell thermocouple numbers below surface (less than 10)
48
49
32
Variables
in program
Comments
Line No.
in file
xshlt(i)
fsolid
Shell thermocouple position below surface
fraction solid for solid temperature location
It is employed to calculate shell thickness:
txshell=(1.-fsolid)*tliq+fsolid*tsol
33
51
52
Section (3) steel properties:
Variables
in program
Comments
Line No.
in file
The compositions of the steel in line 52- 55 are used to find
steel liquidus and solidus temperatures and phase fractions for TLE.
pc pmn ps pp psi
pcr pni pcu pmo pti
pal pv pn pnb pw
pco
icp
isegflag1
CR
%C %Mn %S %P %Si
%Cr %Ni %Cu %Mo %Ti
%Al %V %N %Nb %W
%Co
55
56
57
58
flag of steel grade (1000: carbon steels,
304, 316, 317, 347, 410, 419, 420, 430: AISI stainless steels,
999: user subroutine) Determines which temperaturedependent properties will be used as default in lines 65-72.
59
CK simple Seg. Model flag (1=yes, 0=no)
Cooling rate used in Seg.Model(if above =1) (K/sec)
61
63
In the following lines 65-72, the program will use the constant
property input, or will default to the appropriate grade and
temperature-dependent function if = -1.
slig
ssol
dense
delh
esteel
cpcte
kcte
alpha
steel liquidus temperature (if =-1, then based on grade - see icp)
steel solidus temperature (if =-1, then based on grade - see icp)
steel density (if =-1, then based on grade - see icp)
heat fusion of steel (if =-1, then based on grade - see icp)
steel emissivity (if =-1, then based on grade - see icp)
steel specific heat (if =-1, then based on grade - see icp
steel thermal conductivity (if =-1, then based on grade - see icp
steel thermal expansion coef. (if =-1, then based on grade - see icp)
(otherwise, use for taper calculation to replace phase-fraction TLE
functions)
34
65
66
67
68
69
70
71
Section (4) spray zone variables:
Variables
in program
Comments
tinf
water and ambient temperature in spray zone
75
Line 79-81 choose heat transfer coefficient function: h=ACW^n(1-bT)
Nozaki Model: A*C=0.3925, n=0.55, b=0.0075
Ishiguro Model: A*C=0.581, n=0.451, b=0.0075
coefficient A
78
coefficient n
79
coefficient b
80
minimum convection heat trans. coeff. (natural) (W/m^2K)
81
number of spray zones (for narrow face nzone=1) ( < 30)
82
sprycoefa
sprycoefn
sprycoefb
hnconv0
nzone
Line No.
in file
The table in line 86 - 90 includes the valuables needed
86-90
to define the heat flux in spray zone:
zone start positions, number of rolls in zones, roll radius,
water flow rate, spray zone width, spray zone length,
roll contact angle, fraction of heat flux through roll. i.e. the variables:
izone
dzone
irollsw
rollradw
wflow
sprywidth
sprydist
rollcontact
fracrol
sprycoefc
convect
tambi
izone, dzone, irollsw, rollradw, wflow, sprywidth, sprydist,
rollcontact, fracrol, sprycoefc, convect, tambi (see Figure 3.8)
index of spray cooling zones below the mold
spray zone start points below meniscus
number of rolls in the particular spray zone
roll radius
total flow rate of all nozzles between each pair of rolls in this zone
spray zone width
spry zone length
roll contact angle
fraction of heat flux through the roll
coefficient C in heat transfer coefficient
convect coefficient
ambient temperature for each zone
The end of last spray zone
92
The line numbers that follow assume 5 spray zones, although this is not required.
35
Section (5) mold flux properties:
Variables
in program
Comments
Line No.
in file
line 94 –97 are the compositions of mold flux
wcao wsio2 wmgo wna2o wk2o
%CaO %SiO2 %MgO %Na2O %K2O
wfeo wfe2o3 wnio wmno wcr2o3 %FeO %Fe2O3 %NiO %MnO %Cr2O3
wal2o3 wtio2 wb2o3 wlio2 wsro %Al2O3 %TiO2 %B2O3 %Li2O %SrO
wzro2 wf wfreec wtotc wco2
%ZrO2 %F %free C %total C %CO2
94
95
96
97
ncryst
zcrys
tcrystal
98
100
101
expn
number of Tfsol and viscosity exponent data points
z data (distance down mold below meniscus) (see ncrys)
mold flux solidification temperature (˚C) (see ncrys)
It is used at two places in the program:
(1) to calculate flux viscosity (corresponding to melting point
temperature)
(2) to calculate the thicknesses of liquid flux and solid flux layers
exponent for temperature dependency of viscosity (see ncrys)
102
tksolid
nkl
zkl
tkl
solid flux conductivity
number of distance-Liquid flux conductivity data points
z data (distance down mold below meniscus) (see nkls)
liquid flux conductivity (see nkls)
103
104
106
107
mu
flux viscosity at 1300 ˚C
It is the referent viscosity in flux viscosity calculation
mold flux or slag density (different from powder density)
It is used in continuity equation of mold/shell interface model
flux absorption coefficient
flux index of refraction
slag emissivity
108
rhoflux
acoeff
rindex
eslag
nconsf
cons
form of unit for mold powder consumption rate(see cons)
(1 = kg of flux / m^2)
(2 = kg of flux / tonne of steel, in this case the program will convert
the unit kg/tonne to kg/m^2 according to the thickness and width of
slab, and steel density
mold powder consumption rate, Qf (kg m-2)
36
109
110
111
113
114
115
Variables
in program
Comments
Line No.
in file
zpeak
drim1
drim2
hrim
fluxstrengtht
fluxstrenghtc
poisson
location of peak heat flux (relative to meniscus)
It is used to define the size of the flux rim" size" (the solid flux layer
attached to mold) between the meniscus and peak heat flux locations.
If zpeak =0, the program assumes no rim, i.e. the peak heat flux
is at the meniscus and the input data in lines 82-83 are inactive
slag rim thickness at metal level (meniscus) (read in mm)
slag rim thickness at heat flux peak (read in mm)
Liquid pool depth (read in mm)
Solid flux tensile fracture strength (read in KPa)
Solid flux compress fracture strength (read in KPa)
Solid flux Poisson ratio(-)
117
118
119
120
121
122
nfrc
zfrcs
dfricoeff
fricoeffm
number of distance-Static friction coeff
z data (distance down mold below meniscus) (see nfrc)
Static friction coeff data (see nfrc)
Moving friction coeff data (see nfrc)
123
125
126
126
37
116
Section (6) interface heat transfer variables:
Variables
in program
Comments
nratio
number of distance-ratio data points
( 1=constant ratio of solid flux velocity to casting speed)
distance from meniscus (nratio # of data points)
ratio of average solid flux velocity to casting speed
(nratio # of data points)
134
rrcontact
ecopper
flux/mold contact resistance
mold surface emissivity
136
137
tkair
air conductivity
The program only considers an air gap to exist in oscillation mark
root in the region far below meniscus.
138
ioscflag
flag to decide which oscillation marks model is used
(0=average, 1=transient)
average oscillation mark depth (see Fig 4.1)
It is used to define both volume of liquid flux removed in the marks
and average oscillation mark thickness for interface heat transfer
resistance. Marks are filled with liquid flux/solid flux/air.
width of oscillation mark (see Fig 4.1)
oscillation frequency
(-1 = take default cpm=2*ipm casting speed)
oscillation stroke length (=2.* amplitude)
zratio
vratio
dmark
width
freq
stroke
Line No.
in file
38
130
135
139
140
141
142
144
oscillation mark
Lmark
(width)
dmark
Lpitch
gap
mold
shell
x
z
Fig 4.1 Oscillation marks
39
Section (7) mold water properties:
Variables
in program
Comments
Line No.
in file
The water properties in line 128-131 are used to get the heat
transfer coefficient in the cooling water channel. If the input
data is -1 for these valuables, the program will use default
water properties which vary as a function of temperature.
hwinput
cpwater
rhowater
heat transfer coefficient
(-1 = default = f(T), based on Sleicher and Rouse Eqn)
water heat capacity (-1 = default = f(T))
water density (-1= default = f(T))
40
147
149
150
Section (8) mold geometry:
Variables
in program
Comments
Line No.
in file
dmold1
dmO: mold thickness including water channel (at mold top on outer
153
dmolda
mold face) (see Fig 4.2)
dmI: mold thickness including water channel (at mold top on inner
154
mold face) (see Fig 4.2)
dmoldNF
dwtrEquiv
tNFdiff
Narrow face (NF) mold thickness with water channel (mm)
Equivalent thickness of water box (mm)
Mean temperature diff between hot & cold face of NF (C)
mold side view
dmolda
dmold1
zmen
amrad = RO
meniscus
z
amrada = RI
zmold
copper
mold
outer
face
copper mold
inner face
Fig 4.2 mold outer and inner face
41
155
156
157
Variables
in program
Comments
Line No.
in file
chdepthWF chdepthNF
cooling water channel depth (WF,NF) (see Fig 4.3)
158
chwidthWF chwidthNF
cooling water channel width (WF,NF) (see Fig 4.3)
159
dchannelWF, dchannelNF
distance between cooling water channels (center to center) (WF,NF)
(see Fig 4.3)
160
(for single water channel, set dchannel=chwidth,to make heff=hwater)
totchareaWF totchareaNF
total channel cross sectional area (WF, NF)
(served by water flow line where temp rise measured)
tkmoldWF tkmoldNF
mold thermal conductivity (WF,NF)
alphamold
Mold thermal expansion coeff. (1/K)
twater
cooling water temperature
It is used to calculate cooling water heat transfer coefficient
pwater
cooling water pressure (absolute)
(used only to take adjust water boiling point for warning)
water channel root
bolt
chwidth
hot
steel
region modelled
dchannel
copper mold
twater
bolt
chdepth
xmold
x
y
Fig 4.3 Water channel region
42
mold plating(s)
161
163
164
165
166
Variables
in program
Comments
Line No.
in file
nvwater
unit type for cooling water flowrate/velocity (1=m/s ; 2=L/s)
(L/s refers to volumetric flow rate through each channel)
167
vwaterWF vwaterNF
cooling water flowrate/velocity (WF,NF)
funnelD
funnel height (mm)
funnela
funnel width (mm)
funneltop_b
funnel depth at mold top (mm)
amrada
machine inner radius, RI (see Fig 4.2)
amrad
machine outer radius, RO(see Fig 4.2)
168
171
172
173
174
175
npp
176
number of mold coating/plating thickness changes below
meniscus ( < 20)
The table in line 178- 181 defines the coating / plating
(Ni, Cr, other polynite, and water side scale) thicknesses
at defined distances below meniscus, and their conductivities.
(Each coating is a piece-wise linear function of distance
below meniscus found by connecting the points together)
i.e. the variables:
ip
dscale
dni,
dcr
dpoly
dairg
zpp
akscale
tkni
tkcr
tkpoly
tkair
178-181
ip, dscale, dni, dcr, dpoly, zpp, dairg, akscale, tkni, tkcr, tkpoly, tkairg
Index of data points for the interpolation of coating layer thicknesses
(must be consecutive: 1,2,3,...)
The thickness of water scale at mold cold face
The thickness of Ni coating layer on mold hot face
The thickness of Cr coating layer on mold hot face
The thickness of polynite coating layer on mold hot face
The thickness of air gap
The distance from meniscus
The conductivity of water scale at mold cold face
The conductivity of Ni coating layer on mold hot face
The conductivity of Cr coating layer on mold hot face
The conductivity of polynite coating layer on mold hot face
The conductivity of air
The line numbers that follow assume 3 coating thickness change, although this is not required.
43
Section (9) mold thermocouples:
Variables
in program
nthc
nnthc
xthc
zthc
Comments
the total number of thermocouples ( < 100)
The table in line 180-end defines the thermocouple positions
(for comparing model results with available measurements).
i.e. the variables: nnthc, xthc, zthc
Index of thermocouples
distance beneath mold hot surface
distance below meniscus
44
Line No.
in file
184
187-end
Part V. Nomenclature
Variable
Line #
in input file
Symbol
Standard Value
Unit
acoeff
cpcte
cpwater
dairg
chdepth
xmold
dmold1
dmolda
dmark
dni, dcr, dpoly
tkni, tkcr, tkpoly
dscale
vratio
fsolid
freq
kcte
tkair
tkair
tkliquid
tkmold
akscale
tksolid
tkwaterw
dchannel
width
expn
rindex
bdata
qdata
cons
wflow
amrada
amrad
tinf
98
63
130
153-162
141
a
Cp
Cpwater
dair
dch
dm
dmO
dmI
dmark
dni,dcr,dpoly
kni,kcr,kpoly
dscale
200
690
4179
0
25
m-1
J kg-1 °K-1
J kg-1
mm
mm
56.8
46.8
0.4
0,0,0
80, 72, 4.2
0
0.1
0.7
1.417
30
0.05
0.06
0.8
314.7
0.55
0.8
0.615
29
5
0.85
1.5
0.4
2.0
11.76
12
35
mm
mm
mm
mm
W m-1 °K-1
mm
cycles per second
W m-1 °K-1
W m-1 °K-1
W m-1 °K-1
W m-1 °K-1
W m-1 °K-1
W m-1 °K-1
W m-1 °K-1
W m-1 °K-1
mm
mm
kW m-2
kW m-2
kg m-2
l /min
m
m
°C
134
135
121
153-162
163
153-162
111
48
123
64
119
153-162
95
138
153-162
94
128
141
122
102
99
28
34
104
81-85
149
150
71
fc
fs
freq
k
kair
kair
kliq
km
kscale
ksol
kwater
Lch
Lmark
n
m
qint
qsh
Qf
Qw
RI
RO
T0
45
tcrystal
tinit
twater
vcmmin
yslab
chwidth
dmen
drim1
drim2
zmold
delh
dtime
ecopper
esteel
mu
amuwatw
dense
rhoflux
rhowater
dzone
fflux
fracrol
freqz
i2d
ibound
iseflag
imoldflq
inaro
ioscflag
irollsw
itmax
nbdata
nconsf
nflux
AN
npp
nstep
nthc
93
12
139
11
14
142
136
106
107
15
61
38
118
62
96
129
60
97
131
81-85
29
81-85
40
35
23
67
21
20
120
81-85
46
25
103
31
165
151
39
171
Tfsol
Tpour
Twater
Vc
W
wch
zmen
zrim1
zrim2
zmold
∆HL
∆t
εm
εs
µ0
µwater
ρ
ρflux
ρwater
920
1550
30
1.07
1774
5
100
0
0
810
271
0.02
0.5
0.8
1.1
0.0008
7400
2500
995.6
0, 1 or -1
10
0 or 1
0, 1 or -1
0, or -1
0, 1 or 2
0 or 1
0 or 1
25000
5
50
6
46
°C
°C
°C
m min-1
mm
mm
mm
mm
mm
mm
kJ kg-1
s
poise
Pa-s
kg m3
kg m-3
kg m3
mm
mm
-
nvwater
nzone
pc pmn ps
pp psi
pcr pni
pcu pmo pti
pal pv
pn pnb pw
pco
pwater
rrcontact
RI,RO
rollradw
stroke
submerg
thmax
totcharea
vwater
xshlt
xslabmm
xthc
zana
zbdata
zhbal
zmax
zpeak
zpmin
nnthc
icp
nvc
vctime
slig
ssol
alpha
eslag
nratio
zratio
145
77
51
5
-
52
%
%
53
%
54
140
117
167,168
81-85
125
17
43
143
146
47
13
174-178
37
27
16
42
105
41
174-178
55
7
9
58
59
65
101
111
115
%
MPa
m2 °K W-1
mm
m
mm
mm
mm
mm2
m s-1
mm
mm
mm
mm
mm
mm
mm
m
mm
s
°C
°C
-
0.202
5.e-9
5.e-3, 6.5e-2
10
265
115
7.8
10
230
700
809
811
0
47
wcao wsio2 wmgo
wna2o wk2o
wfeo wfe2o3 wnio
wmno wcr2o3
wal2o3 wtio2
wb2o3 wlio2 wsro
wzro2 wf wfreec
wtotc wco2
zqdata
zthc
89
%
90
%
91
%
92
%
33
174-178
mm
mm
48
Reference
[1] Chandra S., Brimacombe J.K., and Samarasekera I.V. Ironing and Steelmaking, Vol. 20,
1993, n.2, pp.104-112
[2] Nozaki, T., Matsuno, J., Murata, K., Ooi, H., Kodama, M., “A Secondary Cooling Pattern
for Preventing Surface Cracks of Continuous Casting Slab”, Trans. ISIJ, V. 18, 1978,
pp.330-338
[3] J. K. Brimacombe, P.K. Agarwal, S. Hibbins, B.Prabhuker, L.A. Baptista, Spray Cooling in the
Continuous Casting of Steel , Continuous Casting Vol. 2, ISS-AIME, 1984, pp.109-123
49
mass consumption rate of powder flux:
Q(kg/min) =
Q(kg/t) * N W Vc ρ
.46 * .25 * 1 * 1.05 * 7330
=
= 0.89 kg/min
1000
1000
50